Abstract. We prove that every infinite, discrete abelian group admits a pair of $I_{0}$ sets whose union is not $I_{0}$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.

DOI: 10.14232/actasm-016-518-4

AMS Subject Classification (1991): 43A46

Keyword(s): Sidon set, $I_{0}$ set, Kronecker set

Received March 24, 2016, and in revised form August 5, 2016. (Registered under 18/2016.)